Juris Steprans: Universal functions

Universality has various interpretations and it will be the purpose of this talk to establish that many of these are provably different. A graph can be thought of as a function from pairs to 2. It was established by Shelah that it is consistent with the failure of CH that there is a graph of cardinality $\aleph_1$ that is universal for graphs of cardinality $\aleph_1$; in other words, there is a function from pairs of $\omega_1$ to 2 which embeds any other such function. One can use Shelah’s ideas to establish that it is consistent with the failure of CH that there is a function from pairs of $\omega_1$ to $\omega$ that is universal for functions of cardinality $\aleph_1$. It will be shown in this talk that it is consistent that there is a universal function into 2 but not into $\omega$. This is joint work with Shelah.

It will also be shown how to modify this argument to show that it is consistent there is no function from pairs of $\omega_1$ to $\omega$ that is universal for functions of cardinality $\aleph_1$ yet there is such a function of $\omega$ is treated as a set of indiscernibles.